Restocking political capital is a tough business. A trust betrayed can often be a trust destroyed. The Labour leadership is setting out to short-circuit that process by saying sorry, starting with the policy most voters most disliked: immigration. In our interview today, Ed Miliband apologises for the last Labour government being too slow to recognise voters' concerns about the impact of workers from the new EU member countries on the way people live – on pay, jobs, schools and housing – and suggests ways of tackling some of the concerns. In particular, he believes there are ways of limiting the EU's free movement of labour laws by tackling employers who advertise for immigrant workers only, and monitoring industries employing disproportionate numbers of non-UK citizens.

The perception of uncontrolled immigration undoubtedly worries a lot of people, including many who traditionally vote Labour. Mr Miliband is correct that it is an essential issue to address. The problem now is that a lack of political imagination on the left has allowed understandable but damaging myths to get established. It is not clear that Mr Miliband is ready to confront them.

It will certainly be reassuring for many that he promises no repeat of the decision to impose no short-term limit on the number of new EU workers. But he reinforces the claim that these EU migrants took low-skilled jobs and drove down rates of pay, although Jonathan Portes at the National Institute for Economic and Social Research has found find little evidence of a correlation on either point. Mr Miliband is right that this is partly about bad employers; but it is not clear what he believes should be done to restrict access to state support, not least because proportionately fewer migrants than UK workers claim benefits. Nor does he, any more than any other politician, dwell on the evidence that new migrants put much more into the economy than they take from it.

Above all, there is little recognition that the biggest impact of migrant labour is to highlight stress fractures in the national economy. It doesn't look like that, of course, when the doctor's surgery starts putting up signs in Polish, or your child can't get in to the nearest primary school, and the trajectory of the jobless figures seems relentlessly upwards – even when it is not. Mr Miliband is on to an important point about the way the economy works, the "short-term, fast-buck culture". Doing something about that is more complicated. It means tackling the failure to build new housing and the shortage of primary school places. Better employers and tougher enforcement of the rules against the unscrupulous are only part of the solution to low-paid, low-skilled work. This is what people pay taxes for. And taxpayers, of course, include new migrants.